Nonlinear mappings preserving at least one eigenvalue

We prove that if $F$ is a Lipschitz map from the set of all complex $n\times n$ matrices into itself with $F(0)=0$ such that given any $x$ and $y$ we have that $$ and $x-y$ have at least one common eigenvalue, then either $F\left( x\right) =uxu^{-1}$ or $F\left( x\right) =ux^{t}u^{-1}$ for all $x$, for some invertible $n\times n$ matrix $u$. We arrive at the same conclusion by supposing $F$ to be of class $\mathcal{C}$ on a domain in $\mathcal{M}_{n}$ containing the null matrix, instead of Lipschitz. We also prove that if $F$ is of class $\mathcal{C}^{1}$ on a domain containing the null matrix satisfying $F(0)=0$ and $ρ(F\left( x\right) -F\left( y\right) )=ρ(x-y)$ for all $x$ and $y$, where $ρ\left( \cdot \right)$ denotes the spectral radius, then there exists $γ\in \mathbb{C}$ of modulus one such that either $γ^{-1}F$ or $γ^{-1}\overline{F}$ is of the above form, where $\overline{F}$ is the (complex) conjugate of $F$.

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